Integrand size = 9, antiderivative size = 14 \[ \int \sqrt {b x^3} \, dx=\frac {2}{5} x \sqrt {b x^3} \]
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Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 30} \[ \int \sqrt {b x^3} \, dx=\frac {2}{5} x \sqrt {b x^3} \]
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Rule 15
Rule 30
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {b x^3} \int x^{3/2} \, dx}{x^{3/2}} \\ & = \frac {2}{5} x \sqrt {b x^3} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \sqrt {b x^3} \, dx=\frac {2}{5} x \sqrt {b x^3} \]
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Time = 0.11 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.64
method | result | size |
pseudoelliptic | \(\frac {2 x \sqrt {b x}}{3}\) | \(9\) |
gosper | \(\frac {2 x \sqrt {b \,x^{3}}}{5}\) | \(11\) |
default | \(\frac {2 x \sqrt {b \,x^{3}}}{5}\) | \(11\) |
trager | \(\frac {2 x \sqrt {b \,x^{3}}}{5}\) | \(11\) |
risch | \(\frac {2 x \sqrt {b \,x^{3}}}{5}\) | \(11\) |
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Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sqrt {b x^3} \, dx=\frac {2}{5} \, \sqrt {b x^{3}} x \]
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Time = 0.07 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \sqrt {b x^3} \, dx=\frac {2 x \sqrt {b x^{3}}}{5} \]
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Time = 0.22 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sqrt {b x^3} \, dx=\frac {2}{5} \, \sqrt {b x^{3}} x \]
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none
Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \sqrt {b x^3} \, dx=\frac {2}{5} \, \sqrt {b x} x^{2} \mathrm {sgn}\left (x\right ) \]
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Time = 5.81 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.71 \[ \int \sqrt {b x^3} \, dx=\frac {2\,x\,\sqrt {b\,x^3}}{5} \]
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